Endulzar literally means "to sweeten." While it can refer to adding sugar to something, it's often used metaphorically, as it is here.
In "La Despedida," Silvana Estrada uses endulzar la despedida (to sweeten the goodbye) to express the desire to make a difficult farewell less painful and more bearable. It's a beautiful and evocative way to talk about finding comfort and tenderness in a moment of parting.
The whole song is one person trying to postpone a goodbye by relocating it. "Vámonos de aquí" (Let's go from here) opens every verse, and each time it points somewhere else: a place to "endulzar la despedida" (sweeten the farewell), a night where tomorrow can be ignored, a spot "Donde te pueda besar" (Where I can kiss you). Nobody is being asked to stay. The request is only for a better setting to leave from.
The reason arrives in the line that repeats at the close of each half: "Por si un día tú me extrañas, tengas cómo recordar" (In case one day you miss me, you have a way to remember). The kiss, the sea, the pretending are being stockpiled as evidence, so that missing her later has something to hold. By the last verse the singer names the game outright — "Vamos a jugar a fingir que estas volviendo" (Let's play pretend that you are coming back) — while Charlie Hunter's guitar stays low and unhurried underneath, which keeps the whole plea from ever sounding like an argument.